Nonstationarity and Portfolio Choice

NONSTATIONARITY AND PORTFOLIO CHOICE

Christopher B

.

^Barry

Robert L. Winkler

November 1973

Research Reports are p u b l i c a t i o n s reporting on the work o f t h e author. Any views o r conclusions are those o f t h e author, and do n o t n e c e s s a r i l y r e f l e c t those o f IIASA.

Nonstationarity and Portfolio Choice

Christopher B. Barry* and Robert L. Winkler**

Abstract

In this paper some effects of nonstationary para- meters upon inferences and decisions in portfolio anal- ysis are investigated. A Bayesian inferential model with nonstationary parameters is presented and is ap- plied to the problem of portfolio choice. For this model, nonstationarity 1) implies greater uncertainty about future returns; 2) implies that in forecasting future returns, recent returns should receive more weight than not-so-recent returns; 3 ) restricts the amount of information that can be obtained about fu- ture values of the parameters of interest; 4) shifts investment among risky securities and from risky se- curities to risk-free securities; and 5) yields opti- mal portfolios with smaller expected returns than

corresponding optimal portfolios in the stationary case.

1. Introduction

Formal models for decision making under uncertainty generally represent the uncertainty facinq the decision maker in terms of probability distributions. One element of economic reality--change--has been consistently ignored in many areas for which formal models for decision making have been developed. That is, most models have assumed stationary probability distributions in what appears to be a nonstationary world. 1

*

College of Business Administration, University of Florida, U.S.A.

* *

Graduate School of Business, Indiana University, U.S.A.;

research scholar at the International Institute for Applied Systems Analysis.

One area in which formal models have received a great deal of interest but in which change has been persistently overlooked is portfolio analysis. In the work of Markowitz

[2], which laid the foundation for most of the formal port- folio models that are studied today, and in the single-period models that followed (many of which are summarized in Francis and Archer [ 3 ? ) , no mention was made of nonstationarity. In addition, multiperiod models (e.g. Tobin [4]

,

Mossin [5]

,

Chen, Jen, and Zionts [ 6 ] , and Hakansson [7]) have not explic- itly considered the effects of nonstationarity. General

models of consum:>tion and ;rivestment have been developed (e.g.

IIal~-lrsson LL!] Samuelson [9]

,

Fama [lo], and Meyer [ll])

,

and sgain nonstationarity has not been directly confronted.

~ ' I C cco. ~onic arguments for the existence of nonstatio- r.arity in stock price distributions are straightforward. The price behavior of particular securities is affected by char- acteristics of the firm (e.g. capital structure, marketing strategies, product lines, top management, etc.) as well as by the general economic environment in which the firm operates

(e.g. aspects of the econoiny as a whole and characteristics of the competitors, suppliers, and customers of the firm).

As characteristics of a given firm and the economic environ- ment change over time, the anticipated profits of the firm may change, and the valuation of the firm's securities by the stock marlcet may be affected.

Several recent studies ~ r o v i d e empirical support for the claim that the parameters of distributions of stock-price-

r e l a t e d v a r i a b l e s may change o v e r t i m e . I n a s t u d y u s i n g t h e s t a b l e P a r e t i a n model f o r t h e d i s t r i b u t i o n o f l o g p r i c e r e l a - t i v e s , Barnea and Downes [12] found t h a t t h e s t a b l e d i s t r i - b u t i o n s t h a t a p p e a r e d t o e x p l a i n t h e i r d a t a e x h i b i t e d non- s t a t i o n a r y p a r a m e t e r s . P r a e t z [13] found a good f i t f o r s t o c k i n d e x d a t a u s i n g a t d i s t r i b u t i o n , which i s c o n s i s t e n t w i t h r e t u r n s t h a t a r e n o r m a l l y d i s t r i b u t e d w i t h s h i f t i n g v a r - i a n c e , and s t u d y by B l a t t b e r g and Gonedes [14] a m p l i f i e d con- s i d e r a b l y t h e f i n d i n g s of P r a e t z . Boness, Chen, and J a t u s i p i t a k

[15] s u g g e s t e d t h a t l o g p r i c e c h a n g e s a r e n o r m a l l y d i s t r i b u t e d w i t h d i s c r e t e s h i f t s i n v a r i a n c e ; f i r m s were s t u d i e d b e f o r e and a f t e r c a p i t a l s t r u c t u r e c h a n g e s , and t h e l o g p r i c e d i f f e r - e n c e s were found t o have d i f f e r e n t v a r i a n c e s i n t h e two p e r - i o d s . Hsu, M i l l e r , and Wichern [16] o b t a i n e d s i m i l a r r e s u l t s and a r r i v e d a t t h e f o l l o w i n g c o n c l u s i o n [16, p. 1131:

I n a n economy where f a c t o r s a f f e c t i n g s t o c k p r i c e s ( e . g . , t e c h n o l o g i c a l d e v e l o p m e n t , government p o l i c y , e t c . ) a r e s u b j e c t t o d r a - m a t i c s h i f t s , i t d o e s n o t seem s e n s i b l e t o i n s i s t on t h e c o n c e p t of s t a t i o n a r i t y a n d a s i n g l e p r o b a b i l i t y d i s t r i b u t i o n f o r a p r o c e s s g e n e r a t i n g a g i v e n r a t e of r e t u r n s e r i e s .

T h i s p a p e r i s c o n c e r n e d w i t h t h e e f f e c t s o f n o n s t a t i o n - a r i t y on p o r t f o l i o d e c i s i o n s . A c o n v e n i e n t framework f o r s t u d y i n g t h e problem of c h a n g i n g p a r a m e t e r s , b o t h i n t e r m s of f o r e c a s t i n g s e c u r i t y p r i c e s and i n t e r m s of p o r t f o l i o d e c i - s i o n making, i s p r o v i d e d by t h e B a y e s i a n a p p r o a c h t o s t a t i s - t i c a l i n f e r e n c e and d e c i s i o n . Winkler [17] r e c e n t l y d e v e l - oped a B a y e s i a n model f o r f o r e c a s t i n g f u t u r e s e c u r i t y p r i c e s ,

and an e x t e n s i o n o f t h a t model t o t h e n o n s t a t i o n a r y c a s e i s d i s c u s s e d i n S e c t i o n 2 o f t h i s p a p e r . Some r e c e n t r e s u l t s c o n c e r n i n g n o n s - t a t i o n a r y means i n a m u l t i n o r m a l p r o c e s s a r e u t i l i z e d t o i l l u s t r a t e t h e model, and t h e n o n s t a t i o n a r y model i s c o n t r a s t e d w i t h t h e c o r r e s p o n d i n g s t a t i o n a r y model i n t e r m s of b o t h s h o r t - r u n and l i m i t i n g f e a t u r e s . I n S e c t i o n 3 a v e r y b r i e f 2 e s c r i p e i o n of t h e Bayesian p o r t f o l i o s e l e c t i o n and r e v i s i o n mcdel of Winltler and B a r r y el81 i s g i v e n , and i n S e c t i o n 4 t h e r l o n s t a t i o n a r y f o r e c a s t i n g model of S e c t i o n 2 i s u s e d t o d e t e r m i n e t h e e f f e c - t s of n o n s t a t i o n a r i t y on p o r t - f o l i o d e c i s ? . o n s made w i t h t i l e model of S e c t i o n 3 . Two c a s e s a r e examined i n some d e t a i l , t h e c a s e of two r i s k y s e c u r i t i e s 2nd t h e c a s e of one r i s k y s e c u r i t y and one r i s k - f r e e s e c u r i t y .

;,oreo;ler, -the e f f e c t s of n o n s t a t i o n a r i t y on t h e e f f i c i e n t s e t and t h e optimal. p o r t f o l i o from a s t a n d a r d Markowitz-type model a r e a l s o d e s c r i b e d i n S e c t i o n 4 . Some c o n c l u s i o n s and s u g g e s t e d e x t e n s i o n s a r e p r e s e n t e d i n S e c t i o n 5.

2. A B a y e s i a n Model f o r F o r e c a s t i n g S e c u r i t y P r i c e s Under N o n s t a t i o n a r i t y

A

I n % h i s s e c t i o n a model i n v o l v i n g a m u l t i n o r m a l d a t a - generat:ing p r o c e s s w i t h n o n s t a t i o n a r y mean v e c t o r w i l l be c o n s i d e r e d . The random v a r i a b l e s of i n t e r e s t a r e r e t u r n v e c t o r s

it

c o n s i s t i n g of e l e m e n t s

ti,

t h e r e t u r n on s e c u r i t y j(j = I

,...,

^J) d u r i n g t i m e p e r i o d t ( t = 1 , 2

,...

^{) . 2} Suppose t h a t

gt

i s n o r m a l l y d i s t r i b u t e d w i t h unknown mean v e c t o r

ct

and known p o s i t i v e - d e f i n i t e c o v a r i a n c e m a t r i x

--

^C. Assume t h a t a t t h e s t a r t o f p e r i o d t ( i . e . , a t t i m e t - l ) , t h e d e c i s i o n

maker has a normal prior distribution for

6

with mean vector -t

m;

and covariance matrix (n;)-'Z. The marginal distribution of Yt at the start of period t (called a predictive distribu-

-

tion) is then normal with mean vector m; and covariance matrix

-

[(n;

+

^{1) /nil} C . For many decision-making purposes this dis- tribution may be the most essential product of the analysis, and this will be the case in the application of the Bayesian model to portfolio analysis.

After rt is observed, the posterior distribution of

_- ct

is normal3 with mean vector m; and covariance matrix (nl')-l~,

t -

where

and

Next, assume that successive mean vectors satisfy

where is normally distributed with mean vector e and co-

-t

-

variance matrix w-lZ, and

- Et

^and

ct

are independent. Trans- forming the posterior distribution of

Et

into a prior dis- tribution of 6 yields a normal distribution with mean vec-

-t+l tor

and covariance matrix (n

'

⁾ -lC, where t+l

-

(see [20]). In this manner, a sequence of prior and posterior distributions for successive

et

may be obtained as successive F are observed.

-t

Under stationarity,

E

_-t ^{= 0}

_-

for all t, and b

_-

₁ ⁼

c2

^{- -}

...

-

-

-

_ ...

^-

^-

¹

- t! ^-

.

^{Thus, e}

-

^{= 0 and w = 0 ,} in which case (4) and (5) simplify to m' = m" and n' = n'

+

1. Under sta-

-t+l -t t+l t

tionarity, then, the prior distribution of

et+l

at the start of period t-tl is the same as the posterior distribution of

-t at the end of period t. Note that if w-I > 0 , n;

+

^{1 is}

clearly greater than the right-hand side of (5). Thus, n;+l is greater under stationarity than under nonstationarity, given that n; is the same in the two cases.

In the case of nonstationarity with no drift,4 e

-

= 0.

-

Thus, for a given posterior distribution of

ct

at time t, the only difference between the prior distributions of ?t+l under stationarity vis-a-vis nonstationarity with no drift is that the term is larger in the former case.

The covariance matrix of the predictive distribution of at time t is k C, where kt+l = (n;+l

+

^{l)/n;+l. Fur-}

-

^{t+l t+l-}

thermore, an investment portfolio comprised of the J securi- ties may be described by the J x 1 vector a, where the jth

-

element of

a

is the dollar value of investment in security j, and the predictive variance of the value of the portfolio at time t+l is given by the quadratic form T

-

a.

-

^{But kt+l}

is larger under nonstationarity than under stationarity (because is smaller under nonstationarity), and Z is

-

positive definite. Thus, the portfolio variance is greater under nonstationarity, provided that a

- -

# 0, of course. With normal distributions of returns and a risk-averse investor, this implies that any given portfolio is properly viewed as riskier under nonstationarity than under stationarity

.

^This

should not be surprising, since past observations of returns provide relatively less information about the current value of

Et

under nonstationarity than under stationarity. Less information implies greater uncertainty, which should be re- flected by an increase in the measure of uncertainty, variance.

Therefore, considering either a single risky security or a portfolio of securities, nonstationarity implies greater un- certainty. ⁶

Next, consider the limiting behavior of successive prior distributions. In the case of stationary means,

Obviously, as t increases, n' increases without bound. Since t

the covariance matrix of the prior distribution of C(=c

-

^-t for all t) is n 1

-

, it follows that the covariance matrix tends to 0

-.

(a J x J matrix of zeros).

With nonstationary means, successive values of n' are t computed from (2.5)

.

^{For ni > 0,} n; converges to the limit

nL lim n 1 = r(1+4,,,)1/2 -1112

,

t+co t

and t h e convergence i s monotone ( s e e [ 2 0 ] ) . T h e r e f o r e , t h e c o v a r i a n c e m a t r i x of t h e p r i o r d i s t r i b u t i o n of

c

c o n v e r g e s

-t

t o InL) -IT:. T h i s i m p l i e s t h e i n t u i t i v e l y a p p e a l i n g r e s u l t t h a t u n c e r t a i n t y a b o u t

6

c a n n o t be r e d u c e d i n d e f i n i t e l y u n d e r

-t

t h e p r e s c r i b e d c o n d i t i o n s of n o n s t a t i o n a r i t y . I n f a c t , i f n i > nL, i n ' } t w i l l be a d e c r e a s i n g s e q u e n c e , s o t h a t s u c c e s s i v e p r i o r d i s t r i b u t i o n s w i l l r e f l e c t i n c r e a s i n g u n c e r t a i n t y a b o u t t h e i r r e s p e c t i v e arguments

c

The i n a b i l i t y t o e l i m i n a t e

- t '

u n c e r t a i n t y a b o u t

ct

^{a s}

ct

s h i f t s and a d d i t i o n a l r e t u r n s a r e o b s e r v e d i s o f c o u r s e a p r o d u c t o f t h e s t o c h a s t i c n a t u r e o f t h e s h o c k s

C t ,

^{and nL} i s a d e c r e a s i n g f u n c t i o n o f w -1

,

t h e s c a l a r f a c t o r i n t h e c o v a r i a n c e m a t r i x of

- t '

Because t h e s h o c k s a r e s t o c h a s t i c , t h e r e l e v a n c e o f any o b s e r v e d r e t u r n , s a y :t-il f o r making i n f e r e n c e s a b o u t a l a t e r v a l u e o f t h e mean v e c t o r , s a y

iit+l,

d e c r e a s e s a s i i n - c r e a s e s . I n p a r t i c u l a r , t h e w e i g h t a s s i g n e d t o t h e r e t u r n r i n d e t e r m i n i n g a p r i o r mean f o r i s a s t r i c t l y de-

-

^t-i

c r e a s i n g f u n c t i o n of i ( s e e [20]). I n t h e s p e c i a l c a s e i n which n; = nL, f o r example, n; = n f o r a l l t > 1, and t h e

L -

p r i o r mean o f

c

a t t h e s t a r t of p e r i o d t + l c a n be e x p r e s s e d - t + l

i n t h e form

where

Here k i s between z e r o and o n e , and t h e w e i g h t o f :t-i i s (1

-

k ) k i

,

a n e x p o n e n t i a l l y d e c r e a s i n g f u n c t i o n of i. Note t h a t 1

-

k , t h e w e i g h t g i v e n t o t h e most r e c e n t r e t u r n , i s a d e c r e a s i n g f u n c t i o n o f nL ( h e n c e a n i n c r e a s i n g f u n c t i o n o f w -1 ) ; t h e g r e a t e r t h e d e g r e e of n o n s t a t i o n a r i t y , a s measured by t h e s c a l a r f a c t o r of t h e c o v a r i a n c e m a t r i x of

Et,

t h e l e s s w e i g h t i s g i v e n t o a l l b u t t h e most r e c e n t r e t u r n . T h i s p r o - v i d e s a n i n d i c a t i o n of t h e d e c r e a s i n g r e l e v a n c e of a g i v e n p r i c e d i f f e r e n c e i n one p e r i o d f o r making i n f e r e n c e s a b o u t v a l u e s of

ct

f u r t h e r and f u r t h e r i n t o t h e f u t u r e . I n con- t r a s t , a l l of t h e p a s t r e t u r n s a r e w e i g h t e d e q u a l l y u n d e r s t a t i o n a r i t y , and m '

-

^{t + l} can b e e x p r e s s e d i n t h e form

-

n'm

+

^{C ri}

m ' = 1 i=l

-

-t+l

n i + t

One p o s s i b l e e x t e n s i o n of t h e model p r e s e n t e d h e r e would be t h e c o n s i d e r a t i o n of n o n s t a t i o n a r y v a r i a n c e and c o v a r i a n c e t e r m s . To g e n e r a l i z e t h e a p p r o a c h of t h i s s e c t i o n y e t f u r t h e r , r e p l a c e

ct

^by

zt,

which c a n r e p r e s e n t any v a r i - a b l e s of i n t e r e s t , and r e p l a c e

Gt

^by

- s.

The d a t a - g e n e r a t i n g p r o c e s s of i n t e r e s t can t h e n be r e p r e s e n t e d by f

(xt ^{1 9t)} .

^The

p r i o r d i s t r i b u t i o n o f a t t h e s t a r t o f p e r i o d t i s f ' ( Q ~ ) ,

- t

and t h e p o s t e r i o r d i s t r i b u t i o n f " Qt

1

^{x ~ )} c a n be o b t a i n e d once x i s o b s e r v e d :

- t

where fl is the parameter space. At the start of period t, the predictive distribution of

Zt -

may be obtained as follows:

Assume that successive

-

8 are related by -t

where

Ct

is an independent stochastic process with density function g ( ~ ~ )

-

common to all t. (Note that

Ct

is not neces- sarily normally distributed.) In view of (13), the posterior distribution of

8

may be transformed into a prior distribu-

-

-t tion of 8t+l,

-

and successive 0

-

can be dealt with in this fashion. Of -t

course, the tractability of the model depends on the specific distributional assumptions. For example, the normal distri- butions considered earlier in this section yield an especially tractable model.

In this section, it has been demonstrated that the pre- sence of nonstationary means can have an impact upon the un- certainty associated with a given security or set of securi- ties. Moreover, the nonstationary model considered in this section seems to have more realistic properties in this con- text (forecasting security prices) than the corresponding

stationary model. For example, in the nonstationary model the recent returns are given more weight than the not-so-recent returns in determining the mean of the distribution at any given time, and the uncertainty about the parameters of the process is never completely removed (since the covariance matrix of fi does not approach 0 as t increases). Of course,

-t

-

the degree of these effects depends upon the degree of the

-

¹

nonstationarity, as measured by the scalar factor o in the covariance matrix of the shock terms that cause shifts in the mean vector. The model considered here is quite simple, but the properties of the model might be expected to hold under more general conditions, and the same framework can be used to construct forecasting models under much more general con- ditions of nonstationarity.

3. A Bayesian Model for Portfolio Selection and Revision The model discussed in this section is developed and described in greater detail in Winkler and Barry [18]. In this model portfolio management is viewed as an adaptive pro- cess, for sample information is used to update probabilities as more is learned about the properties of the stochastic process generating future security prices. A stationary Bayesian forecasting model is used in [18] to accomplish the probability revision. For simplicity,

F~

will be used in place of mi in this section and the next section to denote the prior

-

mean vector at time t-1 (t = 1,2, ...) of the distribution of

The decision facing the decision maker in the portfolio problem is to determine an optimal portfolio to hold during the first time period. Suppose J securities are under consid- eration for inclusion in the portfolio, and denote the amount invested in security j (j = 1,

...,

J) at time t (t = 0,1,2,. . . )

before portfolio revision by a' Let p: and q: represent the t '

amount of security j that is purchased and sold, respectively, at time t. Thus, the amount invested in security j at time t after revision is a' +

-

^.:q If the rate of return on

t t

security j during period t+l is F!+~, the amount invested in security j at time t+l before revision is

The problem faced by the decision maker at time t (the end of period t) is to choose the vectors p = and

-t

st ^{= {} to maximize E u(Wtel)

,

the expected utility of

-

^j t

= I I:+~, the decision maker's wealth at time t+l.

Wt+l j=1

The subscript on the expectation operator indicates that ex- pectations are taken with respect to the decision maker's prob- ability distribution at time t. It is assumed that trans- actions costs restrict the decision maker's choice of p and

-t

qt.

In particular, if the amount z of a security is bought or sold, a charge of cz is levied. Also, short sales are not allowed, and the amount of a security bought or sold is re- stricted to be nonnegative.

These considerations lead to the following single-period p.,stfolio revision problem:

J

maximize E~ U[

x

⁽¹ + ~:+~)(a: + pt j

_

^gill

j =1 subject to

and

where B = (1

+

^c)/(l

-

^c).

Assume that

gt+l

is normally distributed with unknown mean vector

c -

and known covariance matrix X and that the prior

-

distribution of _.

c

_.. at time t is normal with mean vector m

-

^t+l

-

¹

and covariance matrix

C.

This is the model of Sectlon 2 under stationarity, and the predictive distribution of ?

-t+l is normal with mean vector m _,t+l and covariance matrix kt+l

;.

If the decision maker's utility function for wealth is llnear in wealth, the security i with the highest expected return should be purchased and all those securities j such that 8 (1 t mi+l)

2

(1

+

mt+l) should be sold. That is, the i decision maker should sell those securities for which the transactions costs of selling them and purchasing security i

will be offset by the greater expected return of i.

Under a utility function reflecting risk aversion (i.e.

- dLu ^< O), it is well known that if the distribution of returns dw2

is normal an individual will select a portfolio that is effi- cient in the sense of Markowitz [2]. This implies that the individual will trade off the expected return and variance of various securities in making portfolio selection decisions.

In the portfolio revision model presented in this section, the relevant predictive distribution, ft(~t+l)

,

is normal, so the usual mean-variance tradeof f s are involved. Because of the presence of transactions costs, however, the gains of a par- ticular revision policy must be traded off against the cost of making the shift. In general, the optimal portfolio revi- sion decision under risk aversion will be a function of each security's expected return, the terms of the covariance matrix C , the factor n;+l (which reflects the uncertainty about ii

- -

^t+l

and affects the scale of the predictive covariance matrix), transactions costs, and the amount of each security held in the portfolio prior to revision.

Two utility functions that reflect risk aversion are the quadratic and exponential functions,

and

Under these utility functions, the single-period problem re- duces to very simple forms. In particular, consider the prob- lem with only two securities (J = 2). If any transaction occurs, one security will be bought and one will be sold.

Thus, in view of (17), the amount purchased of one security will be exactly (1

-

^c)/(l

+

^{c) =} 1/B times the amount sold of the other security:

Under the quadratic and exponential utility functions, the single-period portfolio revision problem reduces to the following simple form:

1 2 2 2 1 2

maximize K~ (pt)

+

^{K2 (pt)} + K3 (pt) + K4 (pt) subject to

The coefficients K1, K2, K3, and K4 depend upon the particu- lar utility function that is used, and the exact forms of these coefficients under exponential utility are given in Section 4.

The solution to the quadratic programming problem in C23) is

r

^{( B -1 2 a,, 0)} if -K3/2K1 5 ^{0-I a.:}

(-K3/2~180) if 0 - < -K3/2K1 < B -1 a2 t'

1 2

(pt8 pt) = (0~0) if -K3/2K1 < 0 and -K4/2K2 < 0, (0, - K ~ / ~ K ~ ) if 0 ( -K4/2K2 < B-la:.

-1 1

(0,

B

at) if -K4/2K2

2

^{6-1 a:,}

and t h i s p r o v i d e s t h e o p t i m a l p o r t f o l i o r e v i s i o n p o l i c y f o r t h i s s i t u a t i o n .

I n t h i s s e c t i o n a v e r y b r i e f s k e t c h o f a B a y e s i a n model f o r p o r t f o l i o c h o i c e h a s been p r e s e n t e d . Of c o u r s e , t h e model c o u l d e a s i l y b e g e n e r a l i z e d , and more d e t a i l s a r e g i v e n i n [18]. F o r example, a l t h o u g h o n l y a s i n g l e - p e r i o d model h a s been c o n s i d e r e d h e r e , a m u l t i p e r i o d a d a p t i v e p o r t f o l i o model i s c o n s i d e r e d i n some d e t a i l i n [18].

4 . N o n s t a t i o n a r i t y and P o r t f o l i o Choice

I n S e c t i o n 2 , a Bayesian model f o r f o r e c a s t i n g f u t u r e s e c u r i t y p r i c e s u n d e r n o n s t a t i o n a r y means was p r e s e n t e d , and it was compared w i t h a r e l a t e d s t a t i o n a r y model. The r e s u l t s o f a p p l y i n g t h e s t a t i o n a r y model i n a p o r t f o l i o r e v i s i o n con- t e x t were d e s c r i b e d i n S e c t i o n 3. I n t h i s s e c t i o n , t h e i m p l i - c a t i o n s of n o n s t a t i o n a r i t y f o r p o r t f o l i o c h o i c e w i l l b e con- s i d e r e d t h r o u g h t h e a p p l i c a t i o n o f t h e n o n s t a t i o n a r y f o r e - c a s t i n g model t o t h e p o r t f o l i o r e v i s i o n problem.

I n t h e p o r t f o l i o model of S e c t i o n 3 , t h e p r o b a b i l i t y d i s t r i b u t i o n o f i n t e r e s t a t t i m e t i s t h e p r e d i c t i v e d i s t r i - b u t i o n of ?t+l.

-

I n t h e f o r e c a s t i n g model o f S e c t i o n 2 , t h i s p r e d i c t i v e d i s t r i b u t i o n i s normal w i t h mean v e c t o r m 1

-

^{t + l} and c o v a r i a n c e m a t r i x kt+l

C ,

where kt+l =

+

l)/n;+l. The p r e d i c t i v e d i s t r i b u t i o n u n d e r n o n s t a t i o n a r i t y i s i d e n t i c a l t o t h e d i s t r i b u t i o n t h a t would be o b t a i n e d u n d e r s t a t i o n a r i t y e x c e p t t h a t kt+l i s l a r g e r i n t h e c a s e o f n o n s t a t i o n a r y means t h a n i n t h e c a s e of s t a t i o n a r y means. Hence, s o m e t h i n g c a n b e l e a r n e d a b o u t t h e e f f e c t s o f n o n s t a t i o n a r i t y upon p o r t f o l i o

choice through an analysis of the effects of a change in k t+l upon portfolio choice.

Linear Utility

In the case of linear utility, the optimal portfolio re- vision policy at time t depends only upon m and B . Hence,

-t+l

since the only short-run effect of nonstationarity of the form considered in this section is to change kt+l, decisions made on the basis of linear utility are unaffected by nonstation- arity in the short run.

The limiting case is quite different, however. As indi- cated in Section 2, under stationarity all returns are weight- ed equally and mt converges to

- 6 ^..

as t tends to infinity. Thus, under the decision rule for linear utility, there is only an infinitesimal probability that any security purchases and sales will be made after sufficiently long t because m will

-t

remain virtually unchanged from period to period. Under non- stationarity, however, regardless of how large t is, recent returns are given more weight than not-so-recent returns, and m may change substantially from one period to the next as _-t

Ft

changes. Thus, some purchases and sales may be optimal at any point in time. In effect, the decision maker's evaluation of any security may continue to change regardless of how long the decision maker has been following that security.

Risk Aversion and A Single Risky Security

Nonstationarity has no short-run effect on decisions made via a linear decision rule in a one-period framework, but such is

n o t t h e c a s e when t h e d e c i s i o n maker i s r i s k - a v e r s e . F o r t h e s a k e of i l l u s t r a t i o n , c o n s i d e r p r o b l e m ( 2 3 ) u n d e r e x p o n e n t i a l u t i l i t y . The o p t i m a l p o l i c y i s g i v e n by ( 2 5 ) w i t h

K1 = -dkt+l 2

(01

-

2Bu12 + B 2 u 2 2 ) / 2

,

( 2 6 )

2 2

K2 = -dkt+l ( B u1

-

2BUl2 + 0 2 2 ) / 2

,

( 2 7 )

1 2 1 2

K 3 = (1 + mt+l)

-

B (1 + mt+l)

-

dkt+l [atul

2 1 1 2

+ (at

-

Bat) ul2

-

Bata2

1 ,

( 2 8 )

and

where

1 1

F u r t h e r m o r e , l e t s e c u r i t y one b e r i s k - f r e e ( i . e . mt+l = rt+l and ul 2

= O12 = 0).

From ( 2 5 ) i t i s a p p a r e n t t h a t t h e e f f e c t s of a change i n o n e of t h e p a r a m e t e r s o f t h e p r o b l e m c a n b e a n a l y z e d by con- s i d e r i n g t h e e f f e c t s of t h e c h a n g e upon -K3/2K1 and -K4/2K2.

S i n c e t h e e f f e c t o f n o n s t a t i o n a r i t y i s t o i n c r e a s e k t + l , t h e r e b y i n c r e a s i n g t h e p r e d i c t i v e v a r i a n c e of t h e r e t u r n of t h e r i s k y s e c u r i t y , t h e p a r t i a l d e r i v a t i v e s

and

a r e o f i n t e r e s t . The d e r i v a t i v e i n ( 3 1 ) i s p o s i t i v e u n d e r t h e s u f f i c i e n t ( b u t n o t n e c e s s a r y ) c o n d i t i o n t h a t t h e e x p e c t e d r e t u r n on t h e r i s k y s e c u r i t y e x c e e d s t h a t of t h e r i s k - f r e e s e c u r i t y . The d e r i v a t i v e i n ( 3 2 ) i s n e g a t i v e i f ( 1

+

^mt+l) 2

> B ( 1

+

r t + l ) , which i s a n e c e s s a r y c o n d i t i o n f o r t h e r i s k y 1 s e c u r i t y t o b e p u r c h a s e d . Hence, t h e amount of t h e r i s k y s e c u r i t y h e l d t e n d s t o d e c r e a s e a s n o n s t a t i o n a r i t y i s i n t r o - duced.

Thus, t h e s h o r t - r u n e f f e c t o f n o n s t a t i o n a r i t y i n t h e c a s e of a s i n g l e r i s k y s e c u r i t y competing w i t h a r i s k l e s s a s s e t

(which i s e q u i v a l e n t t o t h e c a s e o f a g i v e n p o r t f o l i o competing w i t h a r i s k l e s s a s s e t ) u n d e r e x p o n e n t i a l u t i l i t y i s t o make t h e r i s k y s e c u r i t y l e s s a t t r a c t i v e . T h i s i s b e c a u s e t h e cen- t r a l e f f e c t of t h e n o n s t a t i o n a r i t y i s t o i n c r e a s e t h e v a r i a n c e of t h e r e t u r n from t h e r i s k y s e c u r i t y w h i l e l e a v i n g i t s ex- p e c t e d r e t u r n unchanged. T h i s r e s u l t a l s o h o l d s i n t h e c a s e of q u a d r a t i c u t i l i t y , and it s h o u l d b e e x p e c t e d t o h o l d f o r any u t i l i t y f u n c t i o n i m p l y i n g a v e r s i o n t o r i s k . The l o n g - r u n e f f e c t o f n o n s t a t i o n a r i t y i n t h e c a s e of a s i n g l e r i s k y s e - c u r i t y and a r i s k - a v e r s e d e c i s i o n maker i s i d e n t i c a l t o t h e

l o n g - r u n e f f e c t i n t h e l i n e a r u t i l i t y c a s e ; p o r t f o l i o r e v i s i o n may be o p t i m a l e v e n f o r l a r g e t , s i n c e t h e d e c i s i o n m a k e r ' s e v a l u a t i o n o f t h e r i s k y s e c u r i t y may c o n t i n u e t o c h a n g e r e g a r d - l e s s of how l o n g t h e d e c i s i o n maker h a s been f o l l o w i n g t h a t s e c u r i t y .

R i s k A v e r s i o n and Two R i s k y S e c u r i t i e s

When t h e two s e c u r i t i e s c o m p e t i n g f o r i n v e s t m e n t d o l l a r s a r e b o t h r i s k y , t h e e f f e c t o f n o n s t a t i o n a r i t y upon an o p t i m a l a l l o c a t i o n s h o u l d b e d e p e n d e n t upon t h e c h a r a c t e r i s t i c s of b o t h of t h e s e c u r i t i e s . Once a g a i n , t h e e f f e c t s o f t h e i n - c r e a s e i n k _{t + l} c a u s e d by n o n s t a t i o n a r i t y c a n b e i n v e s t i g a t e d by f i n d i n g t h e p a r t i a l d e r i v a t i v e s o f -K3/2K1 and -K4/2K2 w i t h r e s p e c t t o kt+l. Moreover, s i n c e b o t h s e c u r i t i e s a r e r i s k y , t h e p r o b l e m i s symmetric, and it i s o n l y n e c e s s a r y t o c o n s i d e r t h e d e r i v a t i v e of -K3/2K The r e s u l t s f o r t h e c a s e of ex-

1'

p o n e n t i a l u t i l i t y a r e p r e s e n t e d h e r e , b a s e d on t h e c o e f f i c i e n t s i n ( 2 6 ) - ( 2 9 ) . ( R e s u l t s u n d e r q u a d r a t i c u t i l i t y , which a r e n o t p r e s e n t e d h e r e , a r e s i m i l a r t o t h e f i n d i n g s u n d e r e x p o n e n t i a l u t i l i t y . )

I n t h e case of e x p o n e n t i a l u t i l i t y ,

S i n c e 0 - > 1 and

I

^{p12 (}

5

1, where p12 = u 12/u1u2 i s t h e c o r - r e l a t i o n c o e f f i c i e n t , o - 2o12

+

0 2 u 2 2 > 0 . Thus, t h e q u a l i -

1

tative effects of nonstationarity in this case, as measured by the sign of the derivative in (331, depend only upon the expected returns of the two securities and upon B (an index reflecting transactions costs), although the actual amounts purchased and sold also depend upon k and 1.

t+l

-

From (33), it follows that if some amount (but not the largest possible amount) of security one should be purchased under stationarity (i.e. if 0 < -K3/2K ₁ ~-'a:)

,

^{then less}

of that security should be purchased under nonstationarity if

more of that security should be purchased if the inequality is reversed, and the same amount should be purchased if equality holds. Given that it is optimal to purchase some of security one, then, the amount purchased increases as the expected re- turn from security one decreases relative to the expected re- turn from security two.* An intuitive explanation can be offered for this somewhat surprising result. For example, suppose that under stationarity some amount of security one should be purchased but that (34) does not hold (i.e. security one is unattractive to a risk-neutral investor). Then in order for a risk-averse individual to buy the security it must offer some reduction in variance for the portfolio. Under nonstationarity the variance of each portfolio is greater, so variance-reduction becomes even more important to the risk- averse investor. Hence, more of the security promising lower portfolio variance should be purchased. On the other hand, if

some amount o f s e c u r i t y one s h o u l d be p u r c h a s e d u n d e r s t a t i o n - a r i t y and ( 3 4 ) h o l d s , t h e n l e s s o f s e c u r i t y one s h o u l d be p u r - c h a s e d u n d e r n o n s t a t i o n a r i t y . T h i s r e s u l t i s i n t u i t i v e l y a p p e a l i n g o n l y i f p u r c h a s i n g s e c u r i t y one i n c r e a s e s t h e p o r t - f o l i o v a r i a n c e ( i . e . i f s e c u r i t y one i s p u r c h a s e d b e c a u s e of a h i g h e x p e c t e d r e t u r n d e s p i t e a h i g h v a r i a n c e ) , which would make s e c u r i t y one l e s s a t t r a c t i v e i n t h e p r e s e n c e of n o n s t a - t i o n a r i t y .

Thus, t h e s h o r t - r u n e f f e c t of n o n s t a t i o n a r i t y i n t h e c a s e of two r i s k y s e c u r i t i e s under e x p o n e n t i a l u t i l i t y depends upon t h e e x p e c t e d r e t u r n s from t h e two s e c u r i t i e s , and t h e s o r t of t r a d e o f f s between e x p e c t e d r e t u r n and v a r i a n c e of r e t u r n t h a t have been d i s c u s s e d i n t h e c o n t e x t of two r i s k y s e c u r i t i e s s h o u l d a l s o h o l d i n problems i n v o l v i n g more t h a n two r i s k y s e c u r i t i e s . Once a g a i n , t h e long-run e f f e c t of n o n s t a t i o n a r i t y i s i d e n t i c a l t o t h e long-run e f f e c t i n t h e l i n e a r u t i l i t y c a s e .

N o n s t a t i o n a r i t y and T r a d i t i o n a l P o r t f o l i o A n a l y s i s The b u l k of s i n g l e - p e r i o d p o r t f o l i o a n a l y s i s i n r e c e n t y e a r s h a s d e a l t w i t h t h e Markowitz p o r t f o l i o s e l e c t i o n model

[2] o r some v a r i a t i o n t h e r e o f . The Markowitz model d e a l s w i t h f i n d i n g an e f f i c i e n t s e t of p o r t f o l i o s ( i . e . t h e s e t o f a l l p o r t f o l i o s h a v i n g minimum v a r i a n c e f o r g i v e n l e v e l s of e x p e c t - e d r e t u r n o r h a v i n g maximum e x p e c t e d r e t u r n f o r g i v e n l e v e l s of v a r i a n c e ) . I n t h i s s e c t i o n t h i s model w i l l b e c o n s i d e r e d i n view o f t h e r e s u l t s i n S e c t i o n 2 c o n c e r n i n g f o r e c a s t i n g u n d e r n o n s t a t i o n a r i t y .

Since the only short-run difference between the stationary and nonstationary models considered here is that the covariance matrix is increased by a scalar factor in the nonstationary case, the effects of nonstationarity may be examined by merely seeing how a scalar increase in the covariance matrix affects the efficient set and the choice of an optimal portfolio. In studying the somewhat related problem of the effects of un- known (but stationary) parameters upon mean-variance portfolio analysis, Barry [ 2 2 ] found that under assumptions similar to those used here, the covariance matrix of returns was multi- plied by a scalar larger than one. Hence, the initial effect was the same as that found here for nonstationarity, and there- fore the analysis in that paper is directly relevant to the current problem.

Nonstationarity leaves the efficient set unchanged in the sense that all portfolios that are efficient under stationarity remain efficient under nonstationarity. However, a map of the efficient set in mean-variance space shows that the set is shifted to the right (following the convention of placing variance on the horizontal axis). This implies that the opti- mal package of risky securities (i.e. the optimal portfolio excluding the risk-free security) changes. The change is to a package with lower expected return, implying a less risky package as well (see [22] for details).

In short, as uncertainty is increased via nonstationarity, the perceived risk of each portfolio increases. This causes the investor to "retreat" to a less risky package. This re- sult is consistent with the earlier findings in this section in

which the effects of nonstationarity were considered in the context of a single-period version of a Bayesian model for portfolio selection and revision.

5. Summary and Discussion

In this paper, a Bayesian model for forecasting future security prices under nonstationarity has been described and compared with a corresponding stationary model. In terms of the short-run behavior of the models, greater uncertainty is retained under nonstationarity than under stationarity. In terms of the limiting behavior of the models, the values of the parameters of interest cannot be ascertained with certainty under nonstationarity, even after the process has been observed for many time periods, and any given observed returns receive less weight as the length of time since the observed returns increases. These properties are not shared by the correspond- ing stationary model, and in general, the nonstationary model considered in this paper appears to have more realistic pro- perties than the corresponding stationary model.

With respect to portfolio choice under linear utility, nonstationarity has no effect in the short run but may prevent the curtailment of trading in the long run that occurs under the stationary model. For a risk-averse decision maker con- sidering one risky security and one risk-free security, non- stationarity decreases the attractiveness of the risky secur- ity. This implies that in general, a risk-averse decision maker will invest less money in a portfolio of risky securi-

ties in the nonstationary case than in the stationary case.

When the two securities under consideration are both risky, the effect of nonstationarity for a risk-averse decision maker can be related to the expected returns for the two securities.

With respect to traditional mean-variance analysis, nonstation- arity does not affect the membership of the efficient set of portfolios, but the efficient set does shift in mean-variance space due to the additional uncertainty under nonstationarity, and this causes a change in the optimal portfolio.

Various extensions of the forecasting model could be con- sidered, and the portfolio selection and revision model could be reexamined in the light of such extensions. In view of recent empirical support for nonstationary variance terms in stock price distributions, the analysis of the effects of non- stationary variances and covariances on portfolio choice would be a logical extension of the analysis in this paper. Winkler [17] considered the case of an unknown covariance matrix, and that approach could be extended to include a nonstationary covariance matrix. Some additional extensions that might add to the realism of the analysis in this paper include the con- sideration of the case in which the process generating changes in the mean vector is characterized by unknown parameters, al- though the model could become quite cumbersome asadditional uncertainty is introduced. (Aleo, as the number of unknown parameters is increased, the identifiability of the model may become a problem, as the data may not permit inferences about all of the parameters of interest.) Another possible exten-

sion is to consider the case in which changes in the unknown parameters occur at random intervals of time rather than at fixed intervals of time. Carter [23] considered such an ex- tension for the univariate situation studied by Bather [ 2 4 ] ,

and it appears to add considerable realism to the model. How- ever, analytical results for that case may be difficult to obtain.

Nonstationarity has long been neglected in the study of economic decision models in general and in the study of port- folio analysis in particular. Although the results of this paper are obtained under a relatively simple model, the point is that nonstationarity can have effects on portfolio decisions and hence upon the functioning of capital markets. Further work of both an empirical and analytical nature concerning the existence of and effects of nonstationarity appears warranted.

Footnotes

l~lthough nonstationarity has been neglected in the study of economic decision models, some recent work in econometrics has dealt with inferential problems of time-varying parameters in regression analysis, and recently a special issue of the Annals of Economic and Social Measurement was devoted exclu- sively to that topic (see Rosenberg [1J for a review of lite-

-

- .

rature in this area).

2~ tilde over a variable indicates that it is a random variable, and vectors and matrices are shown in boldface.

3~ normal prior distribution of is conjugate with re- spect to sampling from a normal data-generating process with unknown mean vector and known covariance matrix

2,

although it is not required that the prior covariance matrix be a scalar multiple of & . See Raiffa and Schlaifer 1191 for a discussion of conjugate families of distributions in Bayesian inference.

4 ~ n the presence of drift, the limiting expected price differences would be infinite, which seems unreasonable. Thus, assuming e

-

= 0

-

appears warranted.

'under these assumptions, variance (or standard deviation) provides a valid measure of risk. For a discussion of this point, see Tobin [2 11

.

6~ecause the prior and posterior distributions of during time period t have been assumed to be identical t for the stationary and nonstationary models, the difference between the values of kt+l in the two models reflects only the impact of a single period of nonstationarity. If several periods are considered, the impact will of course be much greater.

'~ecause of the treatment of as a random variable in the Bayesian forecasting model and because this treatment may be extended to include &, the optimal portfolio will not necessarily-be the same as that derived under the assumption that and _C are known. The qualitative results are similar, but the specifics differ, even if transactions costs are ig- nored (see [22] )

.

8 ~ f course, this is only a "local" effect; as rnt+l de- 1 creases relative to m L t+l, eventually the sign of -K3/2K1 will change and security one will no longer be purchased at all.

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